# extending functor from a dense subcategory

Let $$A$$ and $$B$$ be two cocomplete categories (i.e. closed under small colimits) and $$A'$$ be a dense subcategory of $$A$$ i.e. any object in $$A$$ is a colimit of objects in $$A'$$. Given a functor $$F': A' \to B$$, does there always exist an extension of functor $$F :A \to B$$ preserving all colimits?

• I think you mean "cocomplete", rather than "complete"? And $A'$ is a dense subcategory of $A$, rather than $B$? – Jeremy Rickard Jun 1 at 16:41
• Do you want some condition on $F'$? Otherwise you could just take $A'=A$, with $F'$ some functor that doesn't preserve colimits. – Jeremy Rickard Jun 1 at 16:45
• @JeremyRickard Uhhh, sorry for the typos.. right, I meant cocomplete and $A'$ a dense subcategory of $A$. Maybe I need to say $F'$ preserves colimits. – gregodom Jun 1 at 17:24
• If $A'$ is full in $A$ and skeletally small, then yes: even more is true, there is also an adjunction $[A',Set]\leftrightarrows B$; replace $A'$ with its small skeleton, and use Yoneda lemma. Otherwise, subtle set theory comes in, and I guess you might want to say a bit more on the context of the question :-) – Fosco Jun 1 at 17:29
• arxiv.org/abs/1501.02503 see 3.1.1 here; not because there are no other reference, just because that's the most convenient source to quote for me :) in case $A'$ is not small, see here: ncatlab.org/nlab/show/small+presheaf – Fosco Jun 2 at 18:17

No. This is true only when $$A$$ is a free cocompletion of $$A'$$, i.e., the category of small contravariant functors from $$A$$ to $$Set$$.
• I think I have problem with the condition of the functors being small here. If A′ is small, such condition should be empty. However, in practice (in algebraic geometry), say if we consider A′ be the fppf site over a fixed base scheme S which I think is not really small, there is no category of presheaves because of the size issue. But does the category of sheaves make sense here? if so call it $A$, can we say anything about extending a functor from $A'$ to $A$? – gregodom Jun 2 at 17:45